It suffices to find the amount of time OR the degree measure between two eclipses. In the initial position, there is one eclipse. Fast forward 40 days, and the suns have switched places and the planet has traveled 1/9 of an orbit around the suns. The next moment the suns catch up to the planet, we have another eclipse.

Angular speed = angle / time
Angular speed of suns = 360 deg / 80 days = 9/2 degrees per day
Angular speed of planet = 1 degree / day

Current location of the planet: 40 degrees
Current location of the suns: 0 degrees

If t is the remaining time it will take for the suns to align with the planet, and c is resulting degree measure, then:
0 + (9/2)t = c and 40 + t = c. Therefore,
40 + t = (9/2)t
40 = (7/2)t
(2/7)40 = t
80/7 = t
and (9/2)t = c, so (9/2)(80/7) = 360/7 degrees.
So the angular distance between any two eclipses is 360/7 degrees. Therefore, in one year, there will be seven total eclipses.
Or, the total amount of time between two eclipses is 40 + 80/7 days, or 360/7 days, which in one year will also lead to seven eclipses.